Topics in Delayed Renewal Risk Models

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Abstract

Main focus is to extend the analysis of the ruin related
quantities, such as the surplus immediately prior to ruin, the
deficit at ruin or the ruin probability, to the delayed renewal
risk models.
First, the background for the delayed renewal risk model is
introduced and two important equations that are used as frameworks
are derived. These equations are extended from the ordinary
renewal risk model to the delayed renewal risk model. The first
equation is obtained by conditioning on the first drop below the
initial surplus level, and the second equation by conditioning on
the amount and the time of the first claim.
Then, we consider the deficit at ruin in particular among many
random variables associated with ruin and six main results are
derived. We also explore how the Gerber-Shiu expected discounted
penalty function can be expressed in closed form when
distributional assumptions are given for claim sizes or the time
until the first claim.
Lastly, we consider a model that has premium rate reduced when the
surplus level is above a certain threshold value until it falls
below the threshold value. The amount of the reduction in the
premium rate can also be viewed as a dividend rate paid out from
the original premium rate when the surplus level is above some
threshold value. The constant barrier model is considered as a
special case where the premium rate is reduced to $0$ when the
surplus level reaches a certain threshold value. The dividend
amount paid out during the life of the surplus process until ruin,
discounted to the beginning of the process, is also considered.